International Journal of Advancements in Research & Technology, Volume 2, Issue 8, August-2013
ISSN 2278-7763
244
Image Compression using Singular Value
Decomposition
Miss Samruddhi Kahu
Associate Engineer
Marvell Technologies
Pune, India.
samruddhikahu@gmail.com
Abstract: The Singular Value Decomposition expresses image
data in terms of number of eigen vectors depending upon the
dimension of an image. The psycho visual redundancies in an
image are used for compression. Thus an image can be
compressed without affecting the image quality. This paper
presents one such image compression technique called as SVD.
Basic mathematics of SVD is dealt with in detail and results of
applying SVD on an image are also discussed. The MSE and
compression ratio are used as thresholding, parameters for
reconstruction. SVD is applied on variety of images for
experimentation. The work is concentrated to reduce the number
of eigen values required to reconstruct an image.
Keywords – SVD, Image Compression techniques, Image
Processing.
Ms. Reena Rahate
Professor, Department of Electronics and Communication
Shri Ramdeobaba College of Engg. and Management
Nagpur, India.
reenapalashn@rediffmail.com
maximum. It is observed that if the value of k chosen is equal
to the rank of the image, the reconstructed image is closer to
the original image. As the value of k decreases from the rank
image quality degrades. Second observation is that as the
compression ratio is high image quality is poorer and if
compression ratio is low, image with superior quality can be
reconstructed but with less compression. Therefore
compression ratio and image quality is required to select
appropriately.
Section I presents literature survey and brief outline
of the paper. Section II presents the SVD computations on
images. Section III presents the experimentation and the
proposed procedure to estimate the percentage of eigenvectors
required to reconstruct the original image using the SVD and
PSNR. Section IV presents the results and the discussions.
Finally Section V presents the conclusion.
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I. INTRODUCTION
With the advancement in technology, multimedia content
of digital information is increasing day by day. Which mainly
comprises of images either pictures or video frames. Hence
storage and transmission of these images requires a large
memory space and bandwidth. The solution to this problem is
to reduce the storage space required for these images which
can be done by compressing the image while maintaining
acceptable image quality. Many methods are available for
compression of still images. But the most widely used image
compression technique today is JPEG (Joint Photographic
Experts Group) which uses DCT (Discrete Cosine Transform)
for compression of images. In this paper we are discussing a
image compression technique called SVD (Singular Valued
Decomposition).
Even though DCT gives high energy compaction as
compared to SVD which gives optimal energy compaction,
SVD performs better than DCT in case of images having high
standard deviation (i.e. higher pixel quality).
SVD is a linear matrix transformation used for
compressing images. Using SVD an image matrix is
represented as the product of three matrices U, S, and V where
S is a diagonal matrix whose diagonal entries are singular
values of matrix A. The image A can also be represented by
using less number of singular values, thus, presenting
necessary features of an image while compressing it. The
compressed image requires less storage space as compared to
the original image. To choose the value of k i.e. number of
Eigen values for compression and reconstruction of the image
is an important decision for acceptable reconstruction. It
varies with application and in this work compression ratio is
used to select the number of Eigen values out of the
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II. BASIC MATHEMATICS
Singular Value Decomposition(SVD) is a linear
image matrix transformation in which an image matrix G is
decomposed into 3 component matrices L, D & R such that
G = LDRT
(1)
where G is a m × n matrix
D is a m × n diagonal matrix in which the entries
along the diagonal of D are singular values of G. Singular
value of a matrix is calculated by taking square root of its
eigen value.
L is a m × m matrix containing left singular vectors
of G and
R is a n × n matrix containing right singular vectors
of G.
L and R are orthonormal matrices which means
LLT = I and RRT = I
In matrix form equation (1) can be written as
0
σ 1 0 0………
0 σ 2 0 0 ………….0
G = [l 1 , l 2 ,………,l m] 0 0 σ 3 0 0 0 ……
.
.
.
σr
0 0 ……………. σ N
r1
r2
.
.
.
rn
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International Journal of Advancements in Research & Technology, Volume 2, Issue 8, August-2013
ISSN 2278-7763
Where l 1 , l 2 , …………… l m are m×1 column vectors
r 1 , r 2 , …………... r n are 1×n row vectors and
σ i for i=1, 2,…….N are the singular values of matrix
G.
The singular values are arranged along the
diagonal of D in such a way that
σ 1 ≥σ 2 ≥σ 3 ………..≥σ N ≥0
Also, σ i = √λ i
where λ i for i = 1, 2,……….,N are the eigen values of image
matrix G.
Calculating L, D & R:
From equation (1), we know that
G = LDRT
T
Therefore, G = (LDRT)T = RDTLT
GGT = LDRT × RDTLT
= LDDTLT
= LD2LT
T
(since RR =I)
(2)
From above equation, it is clear that matrix L is obtained
from the eigenvectors of GGT.
Similarly, it can be shown that,
Image compression techniques reduce the number of bits
required to represent an image by taking advantage of these
redundancies. Removing the redundancies is equivalent to
reducing the number of bits required to represent an image
without much compromise in the image quality. Different
image compression techniques apply different methods or
more appropriately coding algorithms to achieve this. Now let
us see how this is achieved using SVD.
By applying SVD on an image, the image matrix G is
decomposed into 3 different matrices L, D and R. However,
simply applying SVD on an image does not compress it.
To compress an image, after applying SVD, only a few
singular values are retained while other singular values are
discarded. This follows from the fact that singular values are
arranged in descending order on the diagonal of D and that
first singular value contains the greatest amount of
information and subsequent singular values contain decreasing
amounts of image information.
Thus, the lower singular values containing negligible or
less important information can be discarded without
significant image distortion.
Furthermore, property 1 of SVD (section 3) says that “the
number of non-zero singular values is equal to the rank of G.”
But even if the lower order singular values after the rank of
the matrix are not zero, they have negligible values and are
treated as noise.
Equation (1) above can also be written as
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GTG = RDTDRT = RD2RT
Therefore, eigenvectors of GTG gives R.
III.
245
PROPERTIES OF SVD
Following are some of the important properties of SVD
1. The rank of matrix G is equal to the number of non-zero
singular values.
2. GGT = LD2LT and hence L daigonalizes the matrix G and
column vectors of L are the eigenvectors of GGT.
3. Similarly, GTG = RD2RT, R diagonalizes G and hence
eigenvectors of GTG are column vectors of R.
4. The singular values of matrix G i.e. σ 1 , σ 2 , σ 3 ……….., σ N
are unique, however the matrices L and R are not unique.
IV. IMAGE COMPRESSION USING SVD
Image compression involves reducing the redundant or
irrelevant information in an image. Redundancies in an image
may be in the form of
1. Psycho visual redundancy, which is due to the
limitations of the human visual system to interpret
very fine details in an image.(i.e. visually nonessential information)
2. Inter pixel redundancy, due to similarities in the
neighboring pixels.
3. Coding redundancy, in which more number of bits
than required are used to encode the image data for
transmission.(i.e. less than optimal code words are
used)
Copyright © 2013 SciResPub.
G=l 1 σ 1 r 1 T + l 2 σ 2 r 2 T ……+ l r σ r r r T ……. +l N σ N r N T
(2)
where r is the rank of G.
From property 1 of SVD (refer section 3 above), it
follows that truncating equation (2) till r values does not make
any significant change in the image. But then the amount of
compression achieved will be very less while the image
quality is nearly same as original.
For good amount of compression to be achieved, only the
first k values of equation (2) are taken so that equation (2)
becomes
G = l 1 σ 1 r 1 T + l 2 σ 2 r 2 T …………+ l k σ k r k T
(3)
where k < r
The image reconstructed using equation (3) above will
reduce the storage space requirement to k*(m+n+1) bytes as
against the storage space requirement of m*n bytes of the
original uncompressed image. Now, our goal of compression
is achieved if the storage space required by the compressed
image is less than that required by the original image. In other
words,
m*n > k*(m+n+1)
(4)
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International Journal of Advancements in Research & Technology, Volume 2, Issue 8, August-2013
ISSN 2278-7763
246
This implies that value of k should be smaller than
m*n/(m+n+1) in order to compress any image thus putting an
upper limit on the value of k.
In short, value of k is chosen such that good amount of
compression is achieved while image quality is maintained
above the minimum acceptable limit.
To compare the results of different compression
techniques and also to measure the degree to which an image
is compressed, many performance measures are available such
as
1. Compression Ratio:
Compression Ratio is the ratio of the storage space
required to store original image to that required to store a
compressed image and is given by
Compression Ratio = m*n / (k*(m+n+1))
It measures the degree to which an image is compressed.
2. Mean Square Error(MSE):
MSE is the measure of deterioration of image quality as
compared to the original image when an image is compressed.
It is defined as square of the difference between pixel value of
original image and the corresponding pixel value of the
compressed image averaged over the entire image.
Mathematically,
MSE = [ ∑ ∑ g(x,y) – g’(x,y) ] / (m*n)
Fig. 1 k = 5
Fig. 2 k = 8
Fig. 4 k = 25
Fig. 3 k = 16
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3. Power Signal to Noise Ratio (PSNR):
As the name suggests, Peak Signal to Noise Ratio
(PSNR) is the ratio of maximum signal power to the noise
power that corrupts it. In Image compression maximum signal
power refers to the original image and noise is introduced to
compress it. In other words, noise is the deviation of the
compressed image from the original one. Therefore, it follows
that PSNR gives the quality of the reconstructed images after
compression. Mathematically, PSNR is given by
PSNR = 10*log 10 [255/√MSE]
Fig. 5 k = 32
Fig. 6 k = 128
V. RESULTS AND DISSCUSSION
Images given below show the results of applying SVD on
frames of video, taking different values of k. fig. 8 shows the
original frame. When the value of k is taken as 5, the image is
very blurred which is shown in fig. 1. k = 5 means that the
image is reconstructed considering only the first five eigen
values of the matrix D. Fig. 2 shows the reconstructed image
with k = 8 which is somewhat less distorted than fig. 1. By
observing figures 1 to 6, it is clear that as the value of k (i.e.
number of Eigen values used for reconstruction of the
compressed image) is increased, the compressed image
approaches the original image. This implies that image quality
goes on increasing as the value of k is increased. When k is
equal to the rank of the image matrix, the reconstructed image
is almost same as the original one. Mean Square Error (MSE)
in this case is very less and PSNR is very high. This can be
observed in fig. 7.
Fig. 7 k = 392
Fig. 8 Original Image
All the above observations can be summarized in the
form of a table as shown below:
TABLE I
SUMMARY OF RESULTS
Value of k
5
8
16
25
32
128
392
Compression
Ratio
50.52
31.577
15.788
10.1046
7.89
1.973
0.644
MSE (dB)
PSNR (dB)
35.192
38.8
35
32.21
30.58
29.047
-79.5
-32.32
-30.2716
-26.478
-23.68
-22.05
-9.2022
240.77
(rank of image)
Copyright © 2013 SciResPub.
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International Journal of Advancements in Research & Technology, Volume 2, Issue 8, August-2013
ISSN 2278-7763
400
405
410
415
0.632
0.624
0.616
0.609
-79.48
-79.47
-79.47
-79.46
247
240.75
240.75
240.74
240.73
In the above table, degree of compression is measured
using compression ratio and MSE & PSNR values expressed
in dB are used as a measure of image quality.
Following conclusions can be drawn on the basis of the
above table 1 and figures (a) to (h):1.
2.
3.
4.
5.
6.
7.
Value of k represents the number of Eigen values
used in the reconstruction of the compressed image.
Smaller the value of k, the more is the compression
ratio (i.e. less storage space is required) but image
quality deteriorates (i.e. larger MSE and smaller
PSNR values).
As the value of k increases, image quality improves
(i.e. smaller MSE and larger PSNR) but more storage
space is required to store the compressed image.
Thus, it is necessary to strike a balance between
storage space required and image quality for good
image compression. And from the above
observations, it is found that optimum compression
results are obtained when MSE of the compressed
image is just less than or equal to 30dB (i.e. MSE ≤
30dB). In our case, this is obtained when value of k is
128.
Generally, choice of k depends on the application.
For instance, in some applications, if image quality is
important then higher values of k are chosen. But
sometimes storage space is more important than
image quality , in that case lower k values are
taken.
But, from equation (4) of section 4 above, it follows
that value of k should be chosen such that, k < m*n/
(m+n+1). Therefore, in our case value of k should be
smaller than 253 in order to achieve compression.
When k is equal to the rank of the image matrix, the
reconstructed image is almost same as the original
one. And as k is increased further, there is very
negligible decrease in MSE and increase in PSNR
values. This means that there is very negligible
improvement in the image quality.
Fig. 9 PSNR (dB) vs Normalized number of Eigen values plotted
on logarithmic scale.
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Fig. 10 MSE(dB) vs Normalized number of Eigen values
All the above results can be graphically expressed as:
Fig. 11 Compression Ratio vs Normalized number of Eigen values
Copyright © 2013 SciResPub.
IJOART
International Journal of Advancements in Research & Technology, Volume 2, Issue 8, August-2013
ISSN 2278-7763
248
VI. CONCLUSIONS
Thus it is observed
that
SVD gives good
compression results with less computational complexity
compared to other compression techniques. A certain degree
of compression as required by an application can be achieved
by choosing an appropriate value of k (i.e. the number of
eigen values). In other words, degree of compression can be
varied by varying the value of k. However to achieve high
value of compression ratio image quality is to be sacrificed
.Therefore it is required to select proper value of k to choose
between compression ratio and image quality. Once the value
of k is selected for specific application or for specific video
the same benchmark can be used for all the frames. Besides
image compression, SVD finds application in noise reduction,
face recognition, water marking, etc.
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